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Courses

This page contains content from the 2007–08 academic year. For current information, visit the Harvard Extension School website at www.extension.harvard.edu.

Mathematics

See courselist below.

For information about related programs and scholarships see:

Math Question Center. For students in MATH E-3, E-8, E-9, E-10, E-15, and E-16, individual help is available at the Math Question Center. The center is open Mondays, 5:30-7:30 pm, and Tuesdays, 7:35-9:35 pm, in Sever Hall 215 beginning the first week of classes each term.

Pedagogical seminars. Graduate-credit students enrolled in courses with weekly lesson study sections must participate in a graduate section for each course in which they are enrolled.

Courselist

Mathematics for Teaching

MATH E-3 Quantitative Reasoning: Practical Math
Course tuition: undergraduate credit $700.

Fall term (12500) (Website) (Printable version): Graeme D. Bird, PhD, Assistant Professor of English, Gordon College. Mondays, 5:30-7:30 pm, Sever Hall, Room 214. Optional sections to be arranged.

Spring term (20389) (Website) (Printable version): Charlotte Bonner, EdM, Mathematics Teacher, Winsor School. Tuesdays, 5:30-7:30 pm, Emerson Hall, Room 108. Optional sections to be arranged.

This course reviews basic arithmetical procedures and their use in everyday mathematics. It also includes an introduction to basic statistics covering such topics as the interpretation of numerical data, graph reading, hypothesis testing, and simple linear regression. The personal computer is utilized for the building of spreadsheets, graphs, and some statistical data analysis. No previous knowledge of these tools is assumed. Recommendations for calculators are made during the first class. Prerequisite: placement test is recommended. (4 credits)

MATH E-8 College Algebra
Course tuition: noncredit and undergraduate credit $700, graduate credit $1,625.

Fall term (10431) (Website) (Printable version): Eric Connally, BA. Thursdays, 6-9 pm, Sever Hall, Room 202. Optional undergraduate-credit sections Tuesdays, 5:30-6:30 pm; required graduate-credit sections Thursdays, 4:45-5:45 pm beginning the second week of classes.

Spring term (20393) (Syllabus) (Printable version): David Abbruzzese, Jr., BSEE. Wednesdays, 5:30-7:30 pm, Sever Hall, Room 214. Optional undergraduate-credit sections to be arranged; required graduate-credit sections Wednesdays, 7:35-8:35 pm, beginning the second week of classes.

This course reviews arithmetic and covers algebraic expressions and equations; their manipulation and use in problem solving; word problems; and an introduction to inequalities, absolute values, and graphing. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: satisfactory placement test score. (4 credits)

MATH E-9 Functions and Graphs: Introduction to Precalculus (22429)
(Website) (Printable version)
Andrew Engelward, PhD, Director, Math for Teaching Program, Harvard Extension School.
Course tuition: noncredit and undergraduate credit $700, graduate credit $1,625.
Spring term: Wednesdays, 5:30-7:30 pm, Emerson Hall, Room 104. Optional undergraduate-credit sections to be arranged; required graduate-credit sections Wednesdays, 7:35-8:35 pm, beginning the second week of classes.

This course introduces students to the language of functions and graphing. We work with polynomial, exponential, and trigonometric functions, all at a fundamental level. The course emphasizes problem solving and applications of math to real-world situations. MATH E-9 is intended for students who have successfully completed a college algebra course such as MATH E-8 but who would like to spend more time reviewing basic precalculus topics involving functions than is possible in MATH E-10. Students interested in taking MATH E-15 should take MATH E-10 after completing MATH E-9. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: MATH E-8, or satisfactory placement test score. (4 credits)

MATH E-10 Precalculus
Course tuition: noncredit and undergraduate credit $700, graduate credit $1,625.

Fall term (12572) (Website) (Printable version): David Arias, EdD, Mathematics Instructor, Andover High School. Mondays, 5:30-7:30 pm, Sever Hall, Room 310. Optional undergraduate-credit sections Tuesdays, 6:30-7:30 pm; required graduate-credit sections Mondays, 7:35-8:35 pm, beginning the second week of classes.

Spring term (22379) (Website) (Printable version): Eric Connally, BA. Thursdays, 6-9 pm, Sever Hall, Room 202. Optional undergraduate-credit sections Tuesdays, 5:30-6:30 pm; required graduate-credit sections Thursdays, 4:45-5:45 pm, beginning the second week of classes.

An intensive course for students with superior algebra skills who want to enroll in MATH E-15 the following term. Requires the use of a graphing calculator. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: A– grade in MATH E-8, B– in MATH E-9, or satisfactory placement test score. (4 credits)

MATH E-15 Introduction to the Calculus A
Course tuition: noncredit and undergraduate credit $700, graduate credit $1,625.

Fall term (10436) (Website) (Printable version): Eric C. Towne, AB. Thursdays, 6-9 pm, Emerson Hall, Room 210. Optional undergraduate-credit sections Mondays, 7:35-8:35 pm; required graduate-credit sections Thursdays, 4:45-5:45 pm beginning the second week of classes.

Spring term (20399) (Website) (Printable version): Eric C. Towne, AB. Thursdays, 6-9 pm, Emerson Hall, Room 210. Optional undergraduate-credit sections Mondays, 7:35-8:35 pm; required graduate-credit sections Thursdays, 4:45-5:45 pm, beginning the second week of classes.

This is a complete course in first-semester calculus. Topics include the meaning, use, and interpretation of the derivative; techniques of differentiation; applications to curve sketching and optimization in a variety of disciplines; the definite integral and some applications; and the fundamental theorem of calculus. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: MATH E-10, or satisfactory placement test score. (4 credits)

MATH E-16 Introduction to the Calculus B and Differential Equations
Course tuition: noncredit and undergraduate credit $700, graduate credit $1,625.

Fall term (10437) (Syllabus) (Printable version): Srdjan Divac, MA. Wednesdays, 7:35-9:35 pm, Sever Hall, Room 103. Optional undergraduate-credit sections Mondays, 7:35-8:35 pm; required graduate-credit sections Wednesdays, 6:30-7:30 pm, beginning the second week of classes.

Spring term (20395) (Syllabus) (Printable version): Srdjan Divac, MA. Wednesdays, 7:35-9:35 pm, Sever Hall, Room 214. Optional undergraduate-credit sections Mondays, 7:35-8:35 pm; required graduate-credit sections Wednesdays, 6:30-7:30 pm, beginning the second week of classes.

This course explores integration and differential equations with applications. It covers most of the topics in a second-semester calculus course but with an emphasis on graphical and numerical work. The use of a graphing calculator with the capability of computing (approximating) definite integrals is required. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisites: MATH E-15, or the equivalent—in other words, an excellent working knowledge of first-semester calculus, including the trigonometric and logarithmic functions—or satisfactory placement test score. (4 credits)

MATH E-21a Multivariable Calculus (11648)
(Website) (Printable version)
Robert Winters, PhD, Lecturer in Mathematics, Brandeis University.
Course tuition: noncredit and undergraduate credit $700, graduate credit $1,625.
Fall term: Thursdays, 7:35-9:35 pm, Harvard Hall, Room 202. Optional sections to be arranged.

This course covers the following topics: calculus of functions of several variables; vectors and vector-valued functions; parameterized curves and surfaces; vector fields; partial derivatives and gradients; optimization; method of Lagrange multipliers; integration over regions in R² and R³; integration over curves and surfaces; Green's theorem, Stokes's theorem, divergence theorem. Graduate-credit students complete additional computer-related or other projects consistent with the course material and their backgrounds. Prerequisites: MATH E-16, or the equivalent and general familiarity with computer use; placement test is recommended. (4 credits)

MATH E-21b Linear Algebra (21474)
(Website) (Printable version)
Robert Winters, PhD, Lecturer in Mathematics, Brandeis University.
Course tuition: noncredit and undergraduate credit $700, graduate credit $1,625.
Spring term: Thursdays, 7:35-9:35 pm, Harvard Hall, Room 202. Optional sections to be arranged.

This course covers the following topics: solving systems of linear equations; matrices and linear transformations; image and kernel of a linear transformation; matrices and coordinates relative to different bases; determinants; eigenvalues and eigenvectors; discrete and continuous dynamical systems; least-squares approximation; applications in function spaces. Graduate-credit students complete additional computer-related or other projects consistent with the course material and their backgrounds. Prerequisites: MATH E-16, or equivalent, and general familiarity with matrix-capable calculators or mathematical software; placement test is recommended. (4 credits)

MATH E-104 Discrete Mathematics with Computer Science Applications (11478)
(Website) (Printable version)
Thomas Koshy, PhD, Professor of Mathematics, Framingham State College.
Course tuition: noncredit and undergraduate credit $700, graduate credit $1,700.
Fall term: Tuesdays, 5:30-7:30 pm, Science Center, Room B-10. Optional sections to be arranged.

This course is an introduction to topics in discrete mathematics that are useful in computer science: mathematical logic, set theory, induction, recursion, combinatorics, relations and functions, and graphs. It includes an introduction to algorithm analysis and techniques for proving correctness of programs. Practical programming examples, as well as proofs of relevant theorems, are discussed. The course is intended both for beginning students of computer science and for computer professionals. Prerequisites: a good working knowledge of Java or another structured language and of precalculus mathematics as evidenced by an appropriate score on the placement test; additional background in computing or in college-level mathematics is desirable but not essential. (4 credits)

Mathematics for Teaching

MATH E-300 Math for Teaching Arithmetic (22822)
(Website) (Printable version)
Bret Benesh, PhD, Preceptor in Mathematics, Harvard University.
Course tuition: noncredit $400, graduate credit $1,625.
Spring term: Wednesdays, 4:30-6:30 pm, Science Center, Room 309.

Why do students have such a difficult time with arithmetic? It could be because arithmetic is much more complex than we initially suspect. For instance, the symbol ½ has at least four different interpretations. When a student encounters this fraction he or she must decide which interpretation will best help solve the problem. In this course, we study why it makes sense to do arithmetic the way we do. Moreover, we see how amazing our number system actually is. (Students should try multiplying 97 by 56 using only Roman numerals if they are skeptical.) This course emphasizes mathematical reasoning rather than mathematical skills and results, and communication is a large part of the course. In addition to the mathematical content, we also discuss how different methods of teaching affect students differently. Prerequisite: familiarity with K-12 mathematics; placement test recommended. (4 credits)

MATH E-301 Math for Teaching Number Theory (12614)
(Website) (Printable version)
Andrew Engelward, PhD, Director, Math for Teaching Program, Harvard Extension School.
Course tuition: noncredit $400, graduate credit $1,625.
Fall term: Wednesdays, 4:30-6:30 pm, English High School, Room 407, 144 McBride Street, Jamaica Plain. Optional sections to be arranged.

There are still many mysteries surrounding number theory and prime numbers as we see in this course. Once considered purely abstract, now number theory is critical for almost all of the security codes used on the Internet. In this course we investigate the makings of basic number systems such as the rational and real numbers, study their properties, and work through the division algorithm, Euclidean algorithm, and modular arithmetic to prove the fundamental theorem of arithmetic. Additional topics may include palindromic numbers, Fibonacci numbers, perfect numbers, numbers in different bases, repeating decimals, and Egyptian fractions. Prerequisite: familiarity with K-12 mathematics; placement test is recommended. (4 credits)

MATH E-302 Math for Teaching Geometry (12119)
(Website) (Printable version)
Andrew Engelward, PhD, Director, Math for Teaching Program, Harvard Extension School.
Course tuition: noncredit $400, graduate credit $1,625.
Fall term: Thursdays, 4:30-6:30 pm, Science Center, Room 309. Optional sections to be arranged.

Geometry is all about symmetry, shape, and space. We begin our exploration by going back to the classic work on geometry, Euclid's The Elements; studying straightedge and compass constructions; and then working our way to more modern topics such as tessellations, Pick's theorem, and fractals. Along the way we also investigate golden rectangles, symmetry groups, constructible numbers, and geometry in higher dimensions. We emphasize mathematical reasoning, and communicating mathematics plays an important role in the course. Prerequisite: familiarity with high school geometry; placement test is recommended. (4 credits)

MATH E-303 Math for Teaching Algebra (22499)
(Syllabus) (Printable version)
Srdjan Divac, MA.
Course tuition: noncredit $400, graduate credit $1,625.
Spring term: Tuesdays, 4:30-6:30 pm, English High School, Room 407, 144 McBride Street, Jamaica Plain. Optional sections Tuesdays, 3:30-4:30 pm, beginning the second week of classes.

This course covers the following topics: elementary algebraic concepts, including set theory; graphs of linear functions; polynomials, including roots of polynomials, factoring, and graphing; an introduction to algebraic structures, including groups and rings; exponentials and logarithms; and measurement, estimation, and problem solving. The course also provides a variety of activities that teachers can bring to their classrooms to enhance student understanding of algebra, graphing, and functions. Prerequisites: knowledge of and experience teaching algebra and precalculus; placement test is recommended. (4 credits)

MATH E-304 Inquiries into Probability and Combinatorics (22438)
(Syllabus) (Printable version)
Matthew Leingang, PhD, Preceptor in Mathematics, Harvard University.
Course tuition: noncredit $400, graduate credit $1,625.
Spring term: Mondays, 4:30-6:30 pm, Science Center, Room 110.

Probability is the study of uncertainty. Beginning with the intuitive frequency interpretation of probability we develop the language necessary to make and decide important questions and resolve puzzling paradoxes. Topics include set theory, sample spaces and events, the fundamental counting principle and basic combinatorics, independent events and conditional probability, random variables, probability distributions, statistical methods, and testing statistical hypotheses. Prerequisite: familiarity with K-12 mathematics; placement test is recommended. (4 credits)

MATH E-309 Teaching Math with Technology: A Critical Examination of the School Curriculum (12979)
(Syllabus) (Printable version)
Benjamin Levy, MAT, Consultant, Mathematics Education and Technology.
Course tuition: noncredit $400, graduate credit $1,625.
Fall term: Mondays, 4:30-6:30 pm, Science Center, Room 110.

This course, for both current and future teachers, examines standard presentations of selected topics from the secondary mathematics curriculum and considers how the presentation of each topic reflects what methods and tools are assumed. Features of graphing calculators, dynamic geometry software, and computer algebra systems are introduced and explored. Teachers write exercises, lessons, and curricula that assume students will make essential use of technology to learn mathematics and to solve problems. Prerequisite: familiarity with secondary school mathematics, from basic algebra and geometry through precalculus; placement test is recommended. (4 credits)

MATH E-311 Analysis: Investigating the Mathematical Process (22428)
(Website) (Printable version)
Andrew Engelward, PhD, Director, Math for Teaching Program, Harvard Extension School.
Course tuition: noncredit $400, graduate credit $1,625.
Spring term: Thursdays, 4:30-6:30 pm, Science Center, Room 309. Optional sections to be arranged.

There is more to mathematics than formulas and procedures. Ever wonder where a theorem comes from or why you should believe it? What does it mean to prove a theorem, and how do mathematicians discover or create proofs? In this course, we begin with basic assumptions (no calculus necessary) and reason our way together until we reach some interesting and sophisticated conclusions, including beautiful results from basic analysis, set theory, and topology. Prerequisites: an interest in puzzles, a solid foundation in precalculus, and the patience to discuss theoretical mathematics as well as an interest in making and critiquing arguments; placement test is recommended. (4 credits)

MATH E-315 Classical Mathematics: Topology and Manifolds (12980)
(Website) (Printable version)
Roberto E. Martínez II, SM.
Course tuition: noncredit $400, graduate credit $1,625.
Fall term: Saturdays, 10 am-noon, Science Center, Room 309. Required sections to be arranged.

How can a coffee cup, a donut, and any other single-holed, compact surface be viewed as manifestations of the same fundamental object? According to classical topology these objects can be deformed into one another by stretching and bending, without changing the number of holes. In this course, we use similar topological methods to study spaces called manifolds, which are more general than ordinary Euclidean space, and to understand the functions that can be defined on them. Quite surprisingly, topology can also be used to answer interesting questions: Are two braided knots equivalent? Are there one-sided, two-dimensional surfaces? What is the fewest number of colors needed to color a map so that adjacent areas have distinct colors? Prerequisites: familiarity with K-12 mathematics and proof methodology; placement test is recommended. (4 credits)

MATH E-316 Classical Mathematics: Complex Analysis and Algebraic Geometry (22873)
(Website) (Printable version)
Roberto E. Martínez II, SM.
Course tuition: noncredit $400, graduate credit $1,625.
Spring term: Saturdays, 10 am-noon, Science Center, Room 309.

Contrary to popular thought, complex numbers came about as a tool to understand certain enigmatic solutions of cubic and quartic equations, not quadratic ones. Finally settled in 1569 in his opus, L'Algebra, Rafael Bombelli introduced the notation "i" for the square root of -1 and formalized its use in algebraic manipulation. Not only did he quell unrest in his mathematical contemporaries but he also ignited an interest that continues today. Little more than two centuries after L'Algebra, Augustin Cauchy developed a calculus of complex functions. This is where our journey begins, and in this course we study algebra, geometry, number theory, analysis, and topology in the complex plane to gain insight into the following questions: How are the prime numbers distributed? How do the singularities of a complex function determine its contour integral value? How are doubly periodic functions naturally related to two dimensional lattices and complex tori? Prerequisites: familiarity with K-12 mathematics and proof methodology; placement test is recommended. (4 credits)